Resonant Response in Nonequilibrium Stationary States Ral - - PowerPoint PPT Presentation

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Introduction Resonant Response Conclusions

Resonant Response in Nonequilibrium Stationary States

Raúl Salgado-García

Department of physics, Facultad de Ciencias Universidad Autónoma del Estado de Morelos

6th International Workshop on Non Equilibrium Thermodynamics and 3rd Lars Onsager Symposium Røros, Norway, 24 August 2012

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

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Introduction Resonant Response Conclusions

Outline

1

Introduction Stationary States Linear Response

2

Resonant Response Linear Response for NESS Relaxation–Response Relation

3

Conclusions

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

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Introduction Resonant Response Conclusions Satationary States Linear Response

Markov Process

H = P2 2m + V(x) + interactions, → dx dt = f(x) + ξ(t), ξ(t) = 0, ξ(t)ξ(t′) = 2βδ(t − t′)

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

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Introduction Resonant Response Conclusions Satationary States Linear Response

Markov Process

H = P2 2m + V(x) + interactions, → dx dt = f(x) + ξ(t), ξ(t) = 0, ξ(t)ξ(t′) = 2βδ(t − t′)

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Satationary States Linear Response

Markov Process

H = P2 2m + V(x) + interactions, → dx dt = f(x) + ξ(t), ξ(t) = 0, ξ(t)ξ(t′) = 2βδ(t − t′)

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

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Introduction Resonant Response Conclusions Satationary States Linear Response

Stationary States

Statistical description : probability density ρ(x, t) Lρ = ∂tρ The observables: e. g. Particle current j(t) = v =

• v(x, t)ρ(x, t)dx

Stationary state: t → ∞ ⇒ ρ(x, t) → Pss LPss = 0,

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Satationary States Linear Response

Stationary States

Statistical description : probability density ρ(x, t) Lρ = ∂tρ The observables: e. g. Particle current j(t) = v =

• v(x, t)ρ(x, t)dx

Stationary state: t → ∞ ⇒ ρ(x, t) → Pss LPss = 0,

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Satationary States Linear Response

Stationary States

Statistical description : probability density ρ(x, t) Lρ = ∂tρ The observables: e. g. Particle current j(t) = v =

• v(x, t)ρ(x, t)dx

Stationary state: t → ∞ ⇒ ρ(x, t) → Pss LPss = 0,

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Satationary States Linear Response

Stationary States

Statistical description : probability density ρ(x, t) Lρ = ∂tρ The observables: e. g. Particle current j(t) = v =

• v(x, t)ρ(x, t)dx

Stationary state: t → ∞ ⇒ ρ(x, t) → Pss LPss = 0,

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Satationary States Linear Response

Stationary States

Statistical description : probability density ρ(x, t) Lρ = ∂tρ The observables: e. g. Particle current j(t) = v =

• v(x, t)ρ(x, t)dx

Stationary state: t → ∞ ⇒ ρ(x, t) → Pss LPss = 0,

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Satationary States Linear Response

Equilibrium stationary states (ESS)

Once the system has reached the stationary state, can we determine whether a system is in equilibrium or not? Detailed balance P(x, t|x′, t′)ρss(x′) = P(x′, t|x, t′)ρss(x)

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

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Introduction Resonant Response Conclusions Satationary States Linear Response

Equilibrium stationary states (ESS)

Once the system has reached the stationary state, can we determine whether a system is in equilibrium or not? Detailed balance P(x, t|x′, t′)ρss(x′) = P(x′, t|x, t′)ρss(x)

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

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Introduction Resonant Response Conclusions Satationary States Linear Response

Non-Equilibrium Stationary States (NESS)

A NESS does not fulfill the detailed balance condition. Generally, the fluxes in a NESS are not zero.

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

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Introduction Resonant Response Conclusions Satationary States Linear Response

Non-Equilibrium Stationary States (NESS)

A NESS does not fulfill the detailed balance condition. Generally, the fluxes in a NESS are not zero.

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

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Introduction Resonant Response Conclusions Satationary States Linear Response

Linear response

∂ρp ∂t = Lρp + εeiωtLpρp ρp ≈ Pss + εeiωtR, R = Response function (L − iω)R = LpPss If we know R we can calculate the response for observables jp = jss + εµ(ω)

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Satationary States Linear Response

Linear response

∂ρp ∂t = Lρp + εeiωtLpρp ρp ≈ Pss + εeiωtR, R = Response function (L − iω)R = LpPss If we know R we can calculate the response for observables jp = jss + εµ(ω)

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Satationary States Linear Response

Linear response

∂ρp ∂t = Lρp + εeiωtLpρp ρp ≈ Pss + εeiωtR, R = Response function (L − iω)R = LpPss If we know R we can calculate the response for observables jp = jss + εµ(ω)

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Satationary States Linear Response

Linear response

∂ρp ∂t = Lρp + εeiωtLpρp ρp ≈ Pss + εeiωtR, R = Response function (L − iω)R = LpPss If we know R we can calculate the response for observables jp = jss + εµ(ω)

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

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Introduction Resonant Response Conclusions Satationary States Linear Response

Einstein Relation and Kubo Formula

Is Pss is a ESS, Einstein Relation: for ω = 0 µ = βD = ∞ v(t0)v(t0+τ)dτ, D = Diffusion coefficient Kubo’s Formula µ(ω) = ∞ v(t0)v(t0 + τ)eiωτdτ,

The Fluctuation-Dissipation Theorem , R. Kubo, Rep. Prog. Phys. 29, 255 (1966). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

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Introduction Resonant Response Conclusions Satationary States Linear Response

Einstein Relation and Kubo Formula

Is Pss is a ESS, Einstein Relation: for ω = 0 µ = βD = ∞ v(t0)v(t0+τ)dτ, D = Diffusion coefficient Kubo’s Formula µ(ω) = ∞ v(t0)v(t0 + τ)eiωτdτ,

The Fluctuation-Dissipation Theorem , R. Kubo, Rep. Prog. Phys. 29, 255 (1966). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Resonant Response

(L − iω)R = LpPss ⇒ R = (iω − L)−1LpPss. For Re[s] < 0 (s − L)−1 = ∞ dt estetL We obtain (s = −δ + iω) R = lim

δ→0+

∞ dt estetLLpPss.

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Resonant Response

(L − iω)R = LpPss ⇒ R = (iω − L)−1LpPss. For Re[s] < 0 (s − L)−1 = ∞ dt estetL We obtain (s = −δ + iω) R = lim

δ→0+

∞ dt estetLLpPss.

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Resonant Response

(L − iω)R = LpPss ⇒ R = (iω − L)−1LpPss. For Re[s] < 0 (s − L)−1 = ∞ dt estetL We obtain (s = −δ + iω) R = lim

δ→0+

∞ dt estetLLpPss.

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Resonant Response

By normalization we have that etLLpPss =

• λ=0

cλeλtPλ where {Pλ}λ is complete basis of eigenfunctions for L, with LPλ = λPλ, λ = 0 ⇒ P0 = Pss The we obtain R =

• λ=0

CλPλ iω − λ

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Resonant Response

By normalization we have that etLLpPss =

• λ=0

cλeλtPλ where {Pλ}λ is complete basis of eigenfunctions for L, with LPλ = λPλ, λ = 0 ⇒ P0 = Pss The we obtain R =

• λ=0

CλPλ iω − λ

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Resonant Response

Proposition: If Pss is an equilibrium state then the spectrum of L is real, i. e. λ ∈ R

The proof is based on a symmetry relation for the Fokker-Plank operator L due to J. Kurchan [J. Phys. A: Math. and Gen. 31, 3719 (1998).]

The amplitude of the response R has a maximum at ω = 0.

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Resonant Response

Proposition: If Pss is an equilibrium state then the spectrum of L is real, i. e. λ ∈ R

The proof is based on a symmetry relation for the Fokker-Plank operator L due to J. Kurchan [J. Phys. A: Math. and Gen. 31, 3719 (1998).]

The amplitude of the response R has a maximum at ω = 0.

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Resonant Response

Proposition: If Pss is an equilibrium state then the spectrum of L is real, i. e. λ ∈ R

The proof is based on a symmetry relation for the Fokker-Plank operator L due to J. Kurchan [J. Phys. A: Math. and Gen. 31, 3719 (1998).]

The amplitude of the response R has a maximum at ω = 0.

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Resonant Response

The eigenvalues of L may be complex only if the stationary states is a NESS. In this case R may have a maximum at some ω = 0. We call this behavior resonant response

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Resonant Response

The eigenvalues of L may be complex only if the stationary states is a NESS. In this case R may have a maximum at some ω = 0. We call this behavior resonant response

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

J = flux operator ⇒ jss = Jss. εeiωtJp = perturbing flux operator jp(t) = J + εeiωtJpp At first order in ε we have µ(ω) := jp − jss ε = Jpss + ∞ JetLLpssdt.

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

J = flux operator ⇒ jss = Jss. εeiωtJp = perturbing flux operator jp(t) = J + εeiωtJpp At first order in ε we have µ(ω) := jp − jss ε = Jpss + ∞ JetLLpssdt.

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

J = flux operator ⇒ jss = Jss. εeiωtJp = perturbing flux operator jp(t) = J + εeiωtJpp At first order in ε we have µ(ω) := jp − jss ε = Jpss + ∞ JetLLpssdt.

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

J = flux operator ⇒ jss = Jss. εeiωtJp = perturbing flux operator jp(t) = J + εeiωtJpp At first order in ε we have µ(ω) := jp − jss ε = Jpss + ∞ JetLLpssdt.

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

J = flux operator ⇒ jss = Jss. εeiωtJp = perturbing flux operator jp(t) = J + εeiωtJpp At first order in ε we have µ(ω) := jp − jss ε = Jpss + ∞ JetLLpssdt.

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

etLLpP0 has integral zero. Then, Psp(t) := Pss + γ−1etLLpP0 is properly normalized to 1 independently of the γ value. Then JetLLpss = Jsp(t) − Jss JetLLpss = j∗(t) − jss

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

etLLpP0 has integral zero. Then, Psp(t) := Pss + γ−1etLLpP0 is properly normalized to 1 independently of the γ value. Then JetLLpss = Jsp(t) − Jss JetLLpss = j∗(t) − jss

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

etLLpP0 has integral zero. Then, Psp(t) := Pss + γ−1etLLpP0 is properly normalized to 1 independently of the γ value. Then JetLLpss = Jsp(t) − Jss JetLLpss = j∗(t) − jss

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

etLLpP0 has integral zero. Then, Psp(t) := Pss + γ−1etLLpP0 is properly normalized to 1 independently of the γ value. Then JetLLpss = Jsp(t) − Jss JetLLpss = j∗(t) − jss

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

The frequency-dependent mobility µ(ω) = Jpss + γ−1 ∞ (j∗(t) − jss)eiωtdt j∗(t) = time–dependent average flux which relaxes to a stationary value, j∗(t) → jss when t → ∞, starting with a specific initial conditions Pinitial = Pss + γLpP0

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

The frequency-dependent mobility µ(ω) = Jpss + γ−1 ∞ (j∗(t) − jss)eiωtdt j∗(t) = time–dependent average flux which relaxes to a stationary value, j∗(t) → jss when t → ∞, starting with a specific initial conditions Pinitial = Pss + γLpP0

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Numerical simulation x

An archetypal model: a tilted periodic potential Langevin equation: dx dt = ˜ γf(x) + ξ(t) + εF(t) f(x) = sin(x) + F0, F(t) = sin(ωt)

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Numerical simulation x

An archetypal model: a tilted periodic potential Langevin equation: dx dt = ˜ γf(x) + ξ(t) + εF(t) f(x) = sin(x) + F0, F(t) = sin(ωt)

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

The Fokker-Planck operator L = − ∂ ∂x

• f(x)

− β−1 ∂ ∂x

• The curren operator is J = f(x) + β−1∂x.

The perturbing operator is Lp = −∂x Jp = 1

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

The Fokker-Planck operator L = − ∂ ∂x

• f(x)

− β−1 ∂ ∂x

• The curren operator is J = f(x) + β−1∂x.

The perturbing operator is Lp = −∂x Jp = 1

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Relaxation and Linear Response

The Fokker-Planck operator L = − ∂ ∂x

• f(x)

− β−1 ∂ ∂x

• The curren operator is J = f(x) + β−1∂x.

The perturbing operator is Lp = −∂x Jp = 1

Resonant Response in Nonequilibrium Steady States, R. Salgado-Garcia Physical Review E 85, 051130 (2012). Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Numerical simulation

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Numerical simulation

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Numerical simulation

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Numerical simulation

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions Linear Response for NESS Relaxation–Response Relation

Relaxation–Response Relation

Numerical simulation

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions

Conclusions

If the system is in a NESS, it may show a resonant response at the frequencies given by the imaginary part of the complex eigenvalues. We have shown a relation between the relaxation of the system and its response. We tested these predictions by numerical simulations.

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States

,

Introduction Resonant Response Conclusions

THANKS!

Raúl Salgado-García Resonant Response in Nonequilibrium Stationary States